Find the (i) lengths of major axes, (ii) coordinates of the vertices

    \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\]

Find the (i) lengths of major axes, (ii) coordinates of the vertices

    \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\]

Given:

    \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\]

Divide by

    \[100\]

to both the sides, we get

    \[\frac{25}{100}{{x}^{2}}+\frac{4}{100}{{y}^{2}}=1\]

    \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{25}=1\]

…(i)

Since,

    \[4\text{ }<\text{ }25\]

So, above equation is of the form,

    \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\]

…(ii)

Comparing eq. (i) and (ii), we get

    \[\begin{array}{*{35}{l}}</strong> <strong>   {{a}^{2}}=\text{ }25\text{ }and\text{ }{{b}^{2}}=\text{ }4  \\</strong> <strong>   \Rightarrow a\text{ }=\text{ }\surd 25\text{ }and\text{ }b\text{ }=\text{ }\surd 4  \\</strong> <strong>   \Rightarrow a\text{ }=\text{ }5\text{ }and\text{ }b\text{ }=\text{ }2  \\</strong> <strong>\end{array}\]

(i) To find: Length of major axes

Clearly, a < b, therefore the major axes of the ellipse is along y axes.

∴Length of major axes =

    \[2a\]

 

    \[\begin{array}{*{35}{l}}</sub> <sub>   =\text{ }2\text{ }\times \text{ }5  \\</sub> <sub>   =\text{ }10\text{ }units  \\</sub> <sub>\end{array}\]

(ii) To find: Coordinates of the Vertices

Clearly, a > b

∴ Coordinate of vertices =

    \[\left( 0,\text{ }a \right)\text{ }and\text{ }\left( 0,\text{ }-a \right)\]

    \[=\text{ }\left( 0,\text{ }5 \right)\text{ }and\text{ }\left( 0,\text{ }-5 \right)\]